Let's solve the given expression step by step. The expression provided is:

\[
(x^2 - 2x)^2 - 11(x^2 - 2x) + 24
\]

### Step 1: Let \( y = x^2 - 2x \).

This simplifies the expression to:

\[
y^2 - 11y + 24
\]

### Step 2: Solve the quadratic equation.

We need to solve the quadratic equation \( y^2 - 11y + 24 = 0 \). We do this by factoring the quadratic.

### Step 3: Factorize the quadratic equation.

Find two numbers that multiply to 24 (the constant term) and add to -11 (the coefficient of \( y \)).

The numbers \(-8\) and \(-3\) work because:

\[
-8 \times -3 = 24 \quad \text{and} \quad -8 + (-3) = -11
\]

So, we can factor the quadratic as:

\[
(y - 8)(y - 3) = 0
\]

### Step 4: Solve for \( y \).

Set each factor equal to zero:

\[
y - 8 = 0 \quad \text{or} \quad y - 3 = 0
\]

So:

\[
y = 8 \quad \text{or} \quad y = 3
\]

### Step 5: Substitute back \( y = x^2 - 2x \).

Now, substitute \( y \) back into the equation \( y = x^2 - 2x \):

1. **Case 1:** \( x^2 - 2x = 8 \)

   \[
   x^2 - 2x - 8 = 0
   \]

   We can factor this quadratic as:

   \[
   (x - 4)(x + 2) = 0
   \]

   Thus, \( x = 4 \) or \( x = -2 \).

2. **Case 2:** \( x^2 - 2x = 3 \)

   \[
   x^2 - 2x - 3 = 0
   \]

   This quadratic factors as:

   \[
   (x - 3)(x + 1) = 0
   \]

   Thus, \( x = 3 \) or \( x = -1 \).

### Final Solutions:

The solutions for \( x \) are:

\[
x = 4, \quad x = -2, \quad x = 3, \quad \text{and} \quad x = -1
\]

Would you like to explore any details or have any questions?

Here are 5 related questions:

1. What are the general steps for solving quadratic equations?
2. How can you check if a quadratic equation is factorable?
3. What is the significance of the discriminant in a quadratic equation?
4. Can you solve a quadratic equation using the quadratic formula?
5. How does completing the square method compare to factorization?

**Tip:** When factoring quadratics, always double-check your factors by multiplying them out to ensure they give you the original quadratic expression.

Learn how to solve the expression (x^2 - 2x)^2 - 11(x^2 - 2x) + 24 with a detailed step-by-step solution. This problem covers key algebraic concepts, including quadratic equations and factoring. Suitable for students in Grades 10-12.

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